High Dimensional Expanders, Ramanujan Complexes and Codes

Project Information

HIEXP

Grant agreement ID: 692854

DOI

10.3030/692854

Project closed

EC signature date 18 May 2016

Start date 1 August 2016

End date 31 July 2021

Funded under

• EXCELLENT SCIENCE - European Research Council (ERC)

Total cost

€ 1 592 500,00

EU contribution

€ 1 592 500,00

Coordinated by

THE HEBREW UNIVERSITY OF JERUSALEM
Israel

Periodic Reporting for period 4 - HIEXP (High Dimensional Expanders, Ramanujan Complexes and Codes)

Reporting period: 2021-02-01 to 2021-07-31

Summary of the context and overall objectives of the project

Expander graphs have been an important subject of research in mathematics and computer science in the last five decades. These graphs have found numerous applications in computer science ( Networks, algorithms, error correcting codes and many more) but also in pure mathematics ( groups, number theory, geometry and more).
The goal of the current project is to develop high dimensional theory of expanders. We already have some applications, but the history of expander graphs teaches us that we should expect also more unexpected applications. indeed, already in the first two years of the project we came to a totally unexpected application to non-approximated groups: this is solving a first case of a series of problems which were completely untouchable till recently. We also try to get locally testable codes. If we will succeed this will be a real contribution to computer science: finding such codes is a major ( if not THE major) problem of modern error correcting codes. It represents a new paradigm: up to now all codes obtained were explicit constructions of what is known to exist by random consideration. Local testability is different: random codes are not locally testable!

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

We have established some of the goals, as well as some totally unexpected results which were not mentioned in the proposal, but are of fundamental importance. The main problem we still working on is the problem of locally testable codes.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

We established: cut-off phenomena for Random walks on Ramanujan complexes, We constructed non-approximated groups. we proved first order rigidity for high rank arithmetic lattices.
Th emain thing we hope to do is finding locally testable codes.

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